a bijection between fractional trees and d angulations
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A bijection between fractional trees and d -angulations Marie Albenque and Dominique Poulalhon LIX CNRS Young workshop in arithmetics and combinatorics June, 22th 2011 Albenque & Poulalhon (LIX CNRS) Bijection for d -angulations

  1. A bijection between fractional trees and d -angulations Marie Albenque and Dominique Poulalhon LIX – CNRS Young workshop in arithmetics and combinatorics – June, 22th 2011 Albenque & Poulalhon (LIX – CNRS) Bijection for d -angulations Madrid, June 22th 2011 1 / 14

  2. Definition of planar maps Planar map = planar connected graph embedded properly in the sphere up to a direct homomorphism of the sphere Rooted planar map = an oriented edge is marked. with a planar embedding = the “outer face” is chosen. Albenque & Poulalhon (LIX – CNRS) Bijection for d -angulations Madrid, June 22th 2011 2 / 14

  3. Triangulations, quadrangualations, . . . Faces = connected components of the plane without the edges of the map. Triangulation, quadrangulation, pentagulation, d -angulation, . . . = map whose faces are all of degree 3, 4, 5, d , . . . Girth = length of the shortest cycle. From now on, only d -angulations of girth d . Albenque & Poulalhon (LIX – CNRS) Bijection for d -angulations Madrid, June 22th 2011 3 / 14

  4. Triangulations, quadrangualations, . . . Faces = connected components of the plane without the edges of the map. Triangulation, quadrangulation, pentagulation, d -angulation, . . . = map whose faces are all of degree 3, 4, 5, d , . . . Girth = length of the shortest cycle. From now on, only d -angulations of girth d . Albenque & Poulalhon (LIX – CNRS) Bijection for d -angulations Madrid, June 22th 2011 3 / 14

  5. Triangulations, quadrangualations, . . . Faces = connected components of the plane without the edges of the map. Triangulation, quadrangulation, pentagulation, d -angulation, . . . = map whose faces are all of degree 3, 4, 5, d , . . . Girth = length of the shortest cycle. From now on, only d -angulations of girth d . Albenque & Poulalhon (LIX – CNRS) Bijection for d -angulations Madrid, June 22th 2011 3 / 14

  6. Enumeration One of the main question when studying some families of maps : How many maps belong to this family ? Tutte ’60s: recursive decomposition Matrix integrals: t’Hooft ’74,Brézin, Itzykson, Parisi and Zuber ’78 , Representation of the symmetric group: Goulden and Jackson ’87 , Bijective approach with labeled trees: Cori-Vauquelin ’81, Schaeffer ’98, Bouttier, Di Francesco and Guitter ’04, Bernardi, Chapuy, Fusy, Miermont, . . . Bijective approach with blossoming trees: Schaeffer ’98, Schaeffer and Bousquet-Mélou ’00, Poulalhon and Schaeffer ’05, Fusy, Poulalhon and Schaeffer ’06. Albenque & Poulalhon (LIX – CNRS) Bijection for d -angulations Madrid, June 22th 2011 4 / 14

  7. Rooted simple triangulations The number of rooted simple triangulations with 2 n faces, 3 n edges and n + 2 vertices is equal to: � 4 n − 2 � n !( 3 n − 1 )! = 1 2 ( 4 n − 3 )! 2 . n · ( 4 n − 2 ) n − 1 � �� � number of blossoming trees with n nodes Blossoming tree = rooted plane tree where each node (= inner vertex) carries exactly two leaves. Theorem (Poulalhon and Schaeffer ’05) There exists a one-to-one correspondence between the set of balanced plane trees with n nodes and two leaves adjacent to each node, and the set of rooted simple triangulations of size n. Albenque & Poulalhon (LIX – CNRS) Bijection for d -angulations Madrid, June 22th 2011 5 / 14

  8. Closure of a blossoming tree Root of the tree is not involved in the local closure ⇒ the tree is balanced. n trees correspond to the same rooted triangulation. Albenque & Poulalhon (LIX – CNRS) Bijection for d -angulations Madrid, June 22th 2011 6 / 14

  9. Closure of a blossoming tree Root of the tree is not involved in the local closure ⇒ the tree is balanced. n trees correspond to the same rooted triangulation. Albenque & Poulalhon (LIX – CNRS) Bijection for d -angulations Madrid, June 22th 2011 6 / 14

  10. Closure of a blossoming tree Root of the tree is not involved in the local closure ⇒ the tree is balanced. n trees correspond to the same rooted triangulation. Albenque & Poulalhon (LIX – CNRS) Bijection for d -angulations Madrid, June 22th 2011 6 / 14

  11. Closure of a blossoming tree Root of the tree is not involved in the local closure ⇒ the tree is balanced. n trees correspond to the same rooted triangulation. Albenque & Poulalhon (LIX – CNRS) Bijection for d -angulations Madrid, June 22th 2011 6 / 14

  12. Closure of a blossoming tree Root of the tree is not involved in the local closure ⇒ the tree is balanced. n trees correspond to the same rooted triangulation. Albenque & Poulalhon (LIX – CNRS) Bijection for d -angulations Madrid, June 22th 2011 6 / 14

  13. Closure of a blossoming tree Root of the tree is not involved in the local closure ⇒ the tree is balanced. n trees correspond to the same rooted triangulation. Albenque & Poulalhon (LIX – CNRS) Bijection for d -angulations Madrid, June 22th 2011 6 / 14

  14. Closure of a blossoming tree Root of the tree is not involved in the local closure ⇒ the tree is balanced. n trees correspond to the same rooted triangulation. Albenque & Poulalhon (LIX – CNRS) Bijection for d -angulations Madrid, June 22th 2011 6 / 14

  15. Closure of a blossoming tree Root of the tree is not involved in the local closure ⇒ the tree is balanced. n trees correspond to the same rooted triangulation. Albenque & Poulalhon (LIX – CNRS) Bijection for d -angulations Madrid, June 22th 2011 6 / 14

  16. Closure of a blossoming tree Root of the tree is not involved in the local closure ⇒ the tree is balanced. n trees correspond to the same rooted triangulation. Albenque & Poulalhon (LIX – CNRS) Bijection for d -angulations Madrid, June 22th 2011 6 / 14

  17. Closure of a blossoming tree Root of the tree is not involved in the local closure ⇒ the tree is balanced. n trees correspond to the same rooted triangulation. Albenque & Poulalhon (LIX – CNRS) Bijection for d -angulations Madrid, June 22th 2011 6 / 14

  18. Closure of a blossoming tree Root of the tree is not involved in the local closure ⇒ the tree is balanced. n trees correspond to the same rooted triangulation. Albenque & Poulalhon (LIX – CNRS) Bijection for d -angulations Madrid, June 22th 2011 6 / 14

  19. Closure of a blossoming tree How to describe the inverse construction ? with orientations. Albenque & Poulalhon (LIX – CNRS) Bijection for d -angulations Madrid, June 22th 2011 6 / 14

  20. Orientations Orientation of a planar map = an orientation is given to each edge We want to consider orientations where the outdegree of each vertex is prescribed → general theory of α -orientation (Felsner). For triangulations: � out ( v ) = 3 for each v not in the root face 3-orientation = out ( v ) = 0 otherwise . Theorem (Schnyder ’89, Felsner ’04) Each rooted triangulation of girth 3 admits a unique minimal 3-orientation, ie. a 3-orientation without counterclockwise cycle. Moreover there exists a directed path from any vertices to the root face : the orientation is accessible. Albenque & Poulalhon (LIX – CNRS) Bijection for d -angulations Madrid, June 22th 2011 7 / 14

  21. Orientations Orientation of a planar map = an orientation is given to each edge We want to consider orientations where the outdegree of each vertex is prescribed → general theory of α -orientation (Felsner). For triangulations: � out ( v ) = 3 for each v not in the root face 3-orientation = out ( v ) = 0 otherwise . Theorem (Schnyder ’89, Felsner ’04) Each rooted triangulation of girth 3 admits a unique minimal 3-orientation, ie. a 3-orientation without counterclockwise cycle. Moreover there exists a directed path from any vertices to the root face : the orientation is accessible. Albenque & Poulalhon (LIX – CNRS) Bijection for d -angulations Madrid, June 22th 2011 7 / 14

  22. Orientations Orientation of a planar map = an orientation is given to each edge We want to consider orientations where the outdegree of each vertex is prescribed → general theory of α -orientation (Felsner). For triangulations: � out ( v ) = 3 for each v not in the root face 3-orientation = out ( v ) = 0 otherwise . Theorem (Schnyder ’89, Felsner ’04) Each rooted triangulation of girth 3 admits a unique minimal 3-orientation, ie. a 3-orientation without counterclockwise cycle. Moreover there exists a directed path from any vertices to the root face : the orientation is accessible. Albenque & Poulalhon (LIX – CNRS) Bijection for d -angulations Madrid, June 22th 2011 7 / 14

  23. Inverse construction Theorem (Poulalhon and Schaeffer ’98) There exists a one-to-one correspondence between the set of balanced plane trees with n nodes and two leaves adjacent to each node, and the set of rooted simple triangulations of size n. Albenque & Poulalhon (LIX – CNRS) Bijection for d -angulations Madrid, June 22th 2011 8 / 14

  24. Inverse construction Theorem (Poulalhon and Schaeffer ’98) There exists a one-to-one correspondence between the set of balanced plane trees with n nodes and two leaves adjacent to each node, and the set of rooted simple triangulations of size n. Albenque & Poulalhon (LIX – CNRS) Bijection for d -angulations Madrid, June 22th 2011 8 / 14

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